I was able to verify that the given z_0 is a singularity, and furthermore an essential singularity. However, I am stumped at how to figure out the residue (value of [itex] a_{-1} [/itex] ) from the given information.

I've tried rearranging letting f(z) be a general Laurent series, and then rearranged the equation. I've tried making multiplicative arguments, like below:

In the past I've been able to figure out the residue from multiplicative arguments and matching powers, but I don't see how I can do that here. Since the given z_0 is an essential singularity, there will necessarily be an infinite number of negative-degree terms, which must cancel with the infinite number of positive-degree terms from the exponential portion, and it gets so messy that I can't resolve anything.

Furthermore, I am troubled by the fact that I would need an infinite number of negative powers to cancel with an infinite number of positive powers, and so I am not even sure if the usual approach of matching powers will terminate or give an answer.

Can anyone provide a hint, or point out an error I may have made?

I should mention that I've found series for the exponential expression and trivially for the 1, both about z0. But I'm not finding a nice pattern for the coefficients of the exponential so that I can equate coefficients. The first few terms I have but they're not pretty

I was able to verify that the given z_0 is a singularity, and furthermore an essential singularity. However, I am stumped at how to figure out the residue (value of [itex] a_{-1} [/itex] ) from the given information.

I've tried rearranging letting f(z) be a general Laurent series, and then rearranged the equation. I've tried making multiplicative arguments, like below:

In the past I've been able to figure out the residue from multiplicative arguments and matching powers, but I don't see how I can do that here. Since the given z_0 is an essential singularity, there will necessarily be an infinite number of negative-degree terms, which must cancel with the infinite number of positive-degree terms from the exponential portion, and it gets so messy that I can't resolve anything.

Furthermore, I am troubled by the fact that I would need an infinite number of negative powers to cancel with an infinite number of positive powers, and so I am not even sure if the usual approach of matching powers will terminate or give an answer.

Can anyone provide a hint, or point out an error I may have made?

I should mention that I've found series for the exponential expression and trivially for the 1, both about z0. But I'm not finding a nice pattern for the coefficients of the exponential so that I can equate coefficients. The first few terms I have but they're not pretty

It doesn't look like an essential singularity to me. It looks like a simple pole. Why do you think it's essential? Did you try just computing the limit z->z0 (z-z0)*f(z)?

Thank you for pointing that out, I must have been careless earlier. Using L'Hopital's rule on the indeterminate "0/0" form, I also now get that it is a simple pole. After that I used a result about Laurent series and residue about a pole of order m.